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An Empirical Investigation of the Black- Scholes Call

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An Empirical Investigation of the Black- Scholes Call International Journal of BRIC Business Research (IJBBR) Volume 6, Number 2, May 2017 AN EMPIRICAL INVESTIGATION OF THE BLACK - SCHOLES CALL OPTION PRICING MODEL WITH REFERENCE TO NSE Rajesh Kumar 1 and Rachna Agrawal 2 1Research Scholar, Department of Management Studies 2Associate Professor, Department of Management Studies YMCA University of Science and Technology, Faridabad, Haryana, India ABSTRACT This paper investigates the efficiency of Black-Scholes model used for valuation of call option contracts written on Eight Indian stocks quoted on NSE. It has been generally observed that the B & S Model misprices options considerably on several occasions and the volatilities are ‘high for options which are highly overpriced. Mispriced worsen with the increased in volatility of the underlying stocks. In most of cases options are also highly underpriced by the model. In this research paper, the theoretical options prices of Nifty stock call options are calculated under the B & S Model. These theoretical prices are compared with the actual quoted prices in the market to gauge the pricing accuracy. KEYWORDS Black-Scholes Model, standard deviation, Mean Error, Root Mean Squared Error, Thiel’s Inequality coefficient etc. 1. INTRODUCTION An effective security market provides three principal opportunities- trading equities, debt securities and derivative products. For the purpose of risk management and trading, the pricing theories of stock options have occupied important place in derivative market. These theories range from relatively undemanding binomial model to more complex B & S Model (1973). The Black-Scholes option pricing model is a landmark in the history of Derivative. This preferred model provides a closed analytical view for the valuation of European style options. An option is a contract, which gives the buyer the right, but not the obligation, to buy or sell specified quantity of the underlying assets, at a strike price on or before the expiration date. The underlying may be physical commodities like wheat, rice, cotton etc. or financial instruments like equity stocks, stock index, bonds etc. There are two types of option- call and put option. The call option gives the buyer the right to buy whereas the put option gives the right to sell. Option allows people to bet on the future events and to reduce the financial risk. There are two kinds of options- American options and European options. The former may be exercised any time before its expiration date while the later can be exercised on its expiration date. DOI :10.14810/ijbbr.2017.6201 1 International Journal of BRIC Business Research (IJBBR) Volume 6, Number 2, May 2017 2. BLACK -SCHOLES MODEL Fisher Black and Myron Scholes (1972) option pricing model is a landmark in the history of financial modeling. This model is used by all worldwide leading stock exchanges, investors and traders. The Black-Scholes model is based on seven assumptions- 1. The asset price follows a Brownian motion with µ and σ constant. 2. There are no transaction costs or taxes. All securities are perfectly divisible. 3. There is no dividend during the life of the derivatives. 4. There are no riskless arbitrage opportunities. 5. Security trading is continuous 6. The risk-free interest rate is constant. 7. The option is exercised at the time of maturity (i.e., it is an European option). The formula for the prices of European call option no dividend paying stocks (Hull, 2008)- -rt C = S.N (d 1) – X.e N (d 2) Where, In(S/X) + (r + 0.5 б2) t d1 =---------------------------- б√ t In(S/X) + (r - 0.5 б2) t d2 =---------------------------- б√ t The variables are- C = Call Price of stock option S = Current Stock price X = Exercise price T = Time remaining until expiration, expressed as a percent of a year r = Current continuously compounded risk-free interest rate б = The instantaneous standard deviation of stock’s return. The unknown parameter of this model is б. The Black-Scholes Model says that the option price, no matter it is call or put, is a function of asset price, time to maturity, exercise price, risk free interest rate and volatility of asset price. All those variables except for the volatility are easily obtainable from the market. б is the only unknown factor in the formula. Б is assumed unchanged while calculating option prices. б is calculated through two approaches historical volatility and implied volatility. The historical volatility is calculated by the annualized standard deviation of historical daily returns. The historical approach is much simpler than the other one. The implied volatility looks more on the future movements. 2 International Journal of BRIC Business Research (IJBBR) Volume 6, Number 2, May 2017 3. REVIEW OF LITERATURE A number of researches have been carried out by the researchers to judge the pricing accuracy of the Black-Scholes model. Blattberg and Gonedes (1974) studied the impact of volatility, as one of the basic parameters, on the option price provided by the Black-Scholes model and concluded that volatility of the underlying stock is stochastic and random. Black (1975) had also identified that this model suffers from the pricing errors. Time to maturity has also effect on the calculation of option prices under the Black-Scholes model. MacBerth and Merville (1979), using implied volatility instead of historical volatility, find that implied volatility is high for in-the-money options but the Black-Scholes model underprices these in-the-money options and secondly implied volatility is low for the out-of-the-money options but this model considerably overprices these out-of-the-money options. Rubinstein (1985) examined the implied volatility on the 30 options classes (Chicago Board of Option Exchange) for a period of two years (from August 1976 to August 1978) and found that the short maturity options had higher implied volatility than long maturity options. However, some researches show that discrepancies between the market option prices and prices calculated under the Black-Scholes model are not large enough to be exploited. LauterBach and Schultz (2012) on pricing warrants, Jordan and Seale (1986) and Blomeyer and Boyd (1988) on futures options written on treasury bond have suggested that there is a very little difference between the market actual price and the Black-Scholes predicted price. Bailey (1987) studied on future option written on gold, Shastri and Tandon (1986) on Future (American options), and Jordan, Seale, McCabe and Kenyon (1987) on futures options written on soyabean have found discrepancies are not enough in the model predicted prices. Varma (2003) studies volatility, using data for a short period of time from June 2001 to February 2002, on Nifty Future and options prices under the Black-Scholes model. He suggests that the volatility is severely mispriced because of the imperfection of the Indian market and market is learning and the impact of learning effects can be seen over a long period of time. Ramazan Gencay and Aslihan Salih (2003) compare the Black-Scholes model against the Feedforward Networks Model using S&P 500 option Index data from January1998 to December 1993. They suggest that the Black-Scholes Model exhibits pricing error at several occasions especially for the deeper out-of-the money options compare to the near out-of-the-money options and this pricing error worsens with increased in volatility. Hence, Feedforward networks provides less pricing error as compare to the Black-Scholes model for the deeper out-of-money options. Rinalini Pathak Kakati (2006) studied the effectiveness of Black-Scholes option pricing model in the Indian context using 2342 call and 1280 put options written from July 2001 to March 2003. She found that the Black-Scholes model misprices options considerably on several occasions. Pricing errors are negative on an average and significantly different from zero. She further suggests that mispricing worsens with both increased in moneyness and increased in the volatility of the stocks. The Black-Scholes model, according to Rinalini Pathak Kakati (2006), overprices short-term options and underprices long-term options. 3 International Journal of BRIC Business Research (IJBBR) Volume 6, Number 2, May 2017 Subrata Kumar Mitra (2008) found that the Black-Scholes model exhibits pricing errors on several parameters. He suggests that the implied volatilities are high for in-the-money options and low for out-of-the-money options indicating that the Black-Scholes model underprices in-the- money options and overprices out-of-the-money options. Mckenzie, Gerace and Sbedar (2007) studied the pricing errors produced by the Black-Scholes model using ASX 200 option Index and suggest that the use of a jump-diffusion approach and implied volatility instead of historical volatility increases the tail properties of the underlying lognormal distribution. Consequently, it increases the pricing accuracy of the Black-Scholes model 1. Rubinstein (1994) studied extensively on implied volatility under the Black-Scholes model for S&P 500 option Index and states that the Black-Scholes model may under price options because the tail properties of underlying lognormal distribution are very small. Emilia Vasile and Dan Armeanu (2009) have worked on the mispricing errors produced by the Black-Scholes Model for pricing options contracts. The operators take into consideration the moneyness of an option and the duration up to the due term thereof, when they calculate the volatility on account of which they evaluate the option. This is a direct consequence of the fact the Black-Scholes model cannot be applied in its original form: the prices of the financial assets do not follow log normal distribution. Tripathi & Gupta (2010) examined the predictive accuracy of the Black-Scholes (BS) model in pricing the Nifty Index option contracts by examining whether the skewness and kurtosis adjusted BS model of Corrado and Su gives better results than the original BS model.
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